If I understand your question, the group formed between the amine and carboxyl group with loss of a water molecule is called an amide. (I tried to parse what you meant by "new groups", and "C=O left from the carboxyl group"). The "C=O left from the carboxyl group" is part of the new group ("amide").

The carbonyl group (C=O) that is left from the carboxyl group is now part of the peptide bond and is called the "amide carbonyl" or "carbonyl group" of the peptide bond. Similarly, the nitrogen atom from the amino group that is now part of the peptide bond is called the "amide nitrogen" or "peptide nitrogen".

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AFAIK they still retain their original names.

You have the Amino group, Carboxyl Group which are alternatively referred to as terminus (N-terminus, C-terminus) if they are at the first and last amino acid in the chain respectively.

Then you have the R group that is also attached to the alpha-carbon of the amino acid. There is a fourth bond on this alpha carbon to a hydrogen.

To answer your question the carboxyl group from the newly added amino acid would (temporarily) be the c-terminus until a new amino acid is added to extend the chain.

If there are other names (I’m sure there are) they aren’t as commonly used.